Let's solve the equation \( 4(2x + 1)^2 + 5 = 0 \).

### Step 1: Isolate the squared term.
Subtract 5 from both sides of the equation:

\[
4(2x + 1)^2 = -5
\]

### Step 2: Divide both sides by 4.
This gives:

\[
(2x + 1)^2 = -\frac{5}{4}
\]

### Step 3: Analyze the equation.
The left side of the equation is a square, and squares of real numbers are always non-negative. Since the right side is negative, the equation has no real solution.

### Conclusion:
There are **no real solutions** to this equation because a square cannot equal a negative number. The equation only has complex solutions.

Would you like to explore complex solutions or need further explanation?

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Here are five related questions to expand your understanding:

1. What is the process for solving quadratic equations with complex solutions?
2. How do you simplify expressions involving square roots of negative numbers?
3. What are the properties of complex numbers?
4. Can you solve similar equations with different constants?
5. How do quadratic equations relate to their graphs on the coordinate plane?

**Tip:** Remember, if an equation involves a square equal to a negative number, the solutions will involve complex numbers!

Explore how to solve the quadratic equation 4(2x + 1)^2 + 5 = 0 with a detailed step-by-step solution. Learn about complex solutions in quadratic equations and the process of isolating squared terms. Suitable for advanced high school students.

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